26 research outputs found
Extended superalgebras from twistor and Killing spinors
The basic first-order differential operators of spin geometry that are Dirac
operator and twistor operator are considered. Special types of spinors defined
from these operators such as twistor spinors and Killing spinors are discussed.
Symmetry operators of massless and massive Dirac equations are introduced and
relevant symmetry operators of twistor spinors and Killing spinors are
constructed from Killing-Yano (KY) and conformal Killing-Yano (CKY) forms in
constant curvature and Einstein manifolds. The squaring map of spinors gives KY
and CKY forms for Killing and twistor spinors respectively. They constitute a
graded Lie algebra structure in some special cases. By using the graded Lie
algebra structure of KY and CKY forms, extended Killing and conformal
superalgebras are constructed in constant curvature and Einstein manifolds.Comment: 7 pages, published versio
Index of Dirac operators and classification of topological insulators
Real and complex Clifford bundles and Dirac operators defined on them are
considered. By using the index theorems of Dirac operators, table of
topological invariants is constructed from the Clifford chessboard. Through the
relations between K-theory groups, Grothendieck groups and symmetric spaces,
the periodic table of topological insulators and superconductors is obtained.
This gives the result that the periodic table of real and complex topological
phases is originated from the Clifford chessboard and index theorems.Comment: 17 pages, published versio
Symmetry operators of Killing spinors and superalgebras in AdS_5
We construct the first-order symmetry operators of Killing spinor equation in
terms of odd Killing-Yano forms. By modifying the Schouten-Nijenhuis bracket of
Killing-Yano forms, we show that the symmetry operators of Killing spinors
close into an algebra in AdS_5 spacetime. Since the symmetry operator algebra
of Killing spinors corresponds to a Jacobi identity in extended Killing
superalgebras, we investigate the possible extensions of Killing superalgebras
to include higher-degree Killing-Yano forms. We found that there is a
superalgebra extension but no Lie superalgebra extension of the Killing
superalgebra constructed out of Killing spinors and odd Killing-Yano forms in
AdS_5 background.Comment: 13 page
Twistor spinors and extended conformal superalgebras
We show that the first-order symmetry operators of twistor spinors can be
constructed from conformal Killing-Yano forms in conformally-flat backgrounds.
We express the conditions on conformal Killing-Yano forms to obtain mutually
commuting symmetry operators of twistor spinors. Conformal superalgebras which
consist of conformal Killing vectors and twistor spinors and play important
roles in supersymmetric field theories in conformal backgrounds are extended to
more general superalgebras by using the graded Lie algebra structure of
conformal Killing-Yano forms and the symmetry operators of twistor spinors. The
even part of the extended conformal superalgebra corresponds to conformal
Killing-Yano forms and the odd part consists of twistor spinors.Comment: 16 pages, published versio
Supergravity backgrounds and symmetry superalgebras
We consider the bosonic sectors of supergravity theories in ten and eleven
dimensions which correspond to the low energy limits of string theories and
M-theory. The solutions of supergravity field equations are known as
supergravity backgrounds and the number of preserved supersymmetries in those
backgrounds are determined by Killing spinors. We provide some examples of
supergravity backgrounds which preserve different fractions of supersymmetry.
An important invariant for the characterization of supergravity backgrounds is
their Killing superalgebras which are constructed out of Killing vectors and
Killing spinors of the background. After constructing Killing superalgebras of
some special supergravity backgrounds, we discuss about the possibilities of
the extensions of these superalgebras to include the higher degree hidden
symmetries of the background.Comment: 11 pages, short review, to appear in Turk. J. Phys. special issue on
General Relativity and Related Topic
Spin Geometry and Some Applications
In this review, basic definitions of spin geometry are given and some of its
applications to supersymmetry, supergravity and condensed matter physics are
summarized. Clifford algebras and spinors are defined and the first-order
differential operators on spinors which lead to the definitions of twistor and
Killing spinors are discussed. Holonomy classification for manifolds admitting
parallel and Killing spinors are given. Killing-Yano and conformal Killing-Yano
forms resulting from the spinor bilinears of Killing and twistor spinors are
introduced and the symmetry operators of special spinor equations are
constructed in terms of them. Spinor bilinears and symmetry operators are used
for constructing the extended superalgebras from twistor and Killing spinors. A
method to obtain harmonic spinors from twistor spinors and potential forms is
given and its implications on finding solutions of the Seiberg-Witten equations
are discussed. Supergravity Killing spinors defined in bosonic supergravity
theories are considered and possible Lie algebra structures satisfied by their
spinor bilinears are examined. Spin raising and lowering operators for massless
field equations with different spins are constructed and the case for
Rarita-Schwinger fields is investigated. The derivation of the periodic table
of topological insulators and superconductors in terms of Clifford chessboard
and index of Dirac operators is summarized.Comment: 70 pages, Based on the lectures given at METU Physics Depertment
between the dates 27 October 2017 and 5 January 201
Transgression field theory at the interface of topological insulators
Topological phases of matter can be classified by using Clifford algebras
through Bott periodicity. We consider effective topological field theories of
quantum Hall systems and topological insulators that are Chern-Simons and BF
field theories. The edge states of these systems are related to the gauge
invariance of the effective actions. For the edge states at the interface of
two topological insulators, transgression field theory is proposed as a gauge
invariant effective action. Transgression actions of Chern-Simons theories for
(2+1)D and (4+1)D and BF theories for (3+1)D are constructed. By using
transgression actions, the edge states are written in terms of the bulk
connections of effective Chern-Simons and BF theories.Comment: 7 pages, title changed, new section, discussions and references
added, published versio
Couplings of gravitational currents with Chern-Simons gravities
The coupling of conserved p-brane currents with non-Abelian gaugetheories is
done consistently by using Chern-Simons forms. Conserved currents localized on
p-branes that have a gravitational origin can be constructed from Killing-Yano
forms of the underlying spacetime. We propose a generalization of the coupling
procedure with Chern-Simons gravities to the case of gravitational conserved
currents. In odd dimensions, the field equations of coupled Chern-Simons
gravities that describe the local curvature on p-branes are obtained. In
special cases of three and five dimensions, the field equations are
investigated in detail.Comment: 8 pages, a new discussion with two paragraphs and some references
added, published versio
Spin raising and lowering operators for Rarita-Schwinger fields
Spin raising and lowering operators for massless field equations constructed
from twistor spinors are considered. Solutions of the spin-
massless Rarita-Schwinger equation from source-free Maxwell fields and twistor
spinors are constructed. It is shown that this construction requires Ricci-flat
backgrounds due to the gauge invariance of the massless Rarita-Schwinger
equation. Constraints to construct spin raising and lowering operators for
Rarita-Schwinger fields are found. Symmetry operators for Rarita-Schwinger
fields via twistor spinors are obtained.Comment: 10 pages, an appendix added, published versio
Generalized symmetry superalgebras
We generalize the symmetry superalgebras of isometries and geometric Killing
spinors on a manifold to include all the hidden symmetries of the manifold
generated by Killing spinors in all dimensions. We show that bilinears of
geometric Killing spinors produce special Killing-Yano and special conformal
Killing-Yano forms. After defining the Lie algebra structure of hidden
symmetries generated by Killing spinors, we construct the symmetry operators as
the generalizations of the Lie derivative on spinor fields. All these
constructions together constitute the structure of generalized symmetry
superalgebras. We examplify the construction on weak and nearly
K\"{a}hler manifolds.Comment: 23 page